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%I #15 Aug 07 2023 03:55:14
%S 1,2,4,5,7,8,10,12,13,15,16,20,23,24,25,28,31,32,36,37,39,40,44,47,48,
%T 52,55,56,58,60,63,64,68,71,72,76,79,80,84,85,87,88,92,95,96,100,103,
%U 104,108,111,112,116,119,120,124,127,128,130,132,135,136,140,143,144
%N Positive numbers that are not representable as a primitive sum of three nonzero squares.
%C This is the complement of A223731. There an F. Halter-Koch reference is given.
%H Alois P. Heinz, <a href="/A223735/b223735.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) has no representation as a^2 + b^2 + c^2 with 0 < a <= b <= c and gcd(a,b,c) = 1.
%F Conjectured g.f.: (2*x^61 -x^60 +2*x^59 -x^58 -2*x^57 +x^43 +3*x^42 -3*x^41 +x^40 -2*x^39 +2*x^32 -x^31 +2*x^30 -x^29 -2*x^28 +x^23 +3*x^22 -3*x^21 +x^20 -2*x^19 +x^18 +2*x^16 -3*x^14 +x^12 +3*x^11 -x^10 +x^6 -x^5 +x^4 +2*x^2 +x +1)*x / (x^4 -x^3 -x +1). - _Alois P. Heinz_, Apr 06 2013
%e For a(1) up to a(7) there is no representation as sum of three nonzero squares.
%e a(8) = 12 because the only representation of 12 as a sum of nonzero squares is given by [a,b,c] = [2,2,2] = 2*[1,1,1], and this is not a primitive sum because gcd(2,2,2) = 2, not 1.
%t notThreeSquaresQ[n_] := Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ] == {}; Select[Range[200], notThreeSquaresQ] (* _Jean-François Alcover_, Jun 21 2013 *)
%Y Cf. A223730, A223731, A223732, A223733, A223734.
%K nonn
%O 1,2
%A _Wolfdieter Lang_, Apr 06 2013
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